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We study the processes e{sup +}e{sup -} {yields} K{sup +}K{sup -}{pi}{sup +}{pi}{sup -}{gamma}, K{sup +}K{sup -}{pi}{sup 0}{pi}{sup 0}{gamma} and K{sup +}K{sup -}K{sup +}K{sup -}{gamma}, where the photon is radiated from the initial state. About 34600, 4400 and 2300 fully reconstructed events respectively, are selected from 232 fb{sup -1} of BABAR data. The invariant mass of the hadronic final state defines the effective e{sup +}e{sup -} center-of-mass energy, so that the K{sup +}K{sup -}{pi}{sup +}{pi}{sup -}{gamma} data can be compared with direct measurements of the e{sup +}e{sup -} {yields} K{sup +}K{sup -}{pi}{sup +}{pi}{sup -}{gamma} reaction; no direct measurements exist for the e{sub +}e{sub -} {yields} K{sup +}K{sup -}{pi}{sup 0}{pi}{sup 0}{gamma} or e{sub +}e{sub -} {yields} K{sup +}K{sup -}K{sup +}K{sup -} reactions. Studying the structure of these events, we find contributions from a number of intermediate states, and we extract their cross sections where possible. In particular, we isolate the contribution from e{sup +}e{sup -} {yields} {phi}(1020)f{sub 0}(980) and study its structure near threshold. In the charmonium region, we observe the J/{psi} in all three final states and several intermediate states, as well as the {psi}/(2S) in some modes, and measure the corresponding branching fractions. We see no signal for the Y(4260) and obtain an upper limit of {Beta}{sub Y(4260){yields}{phi}{pi}{sup +}{pi}{sup -}} {center_dot} {Lambda}{sub ee}{sup Y} &lt; 0.4 eV at 90% C.L.

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